Level-set method

The Level-set method (LSM) is a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes.

LSM can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects.

These characteristics make LSM effective for modeling objects that vary in time, such as an airbag inflating or a drop of oil floating in water.

In the upper left corner is a bounded region with a well-behaved boundary.

determining this shape, and the flat blue region represents the X-Y plane.

It is challenging to describe this transformation numerically by parameterizing the boundary of the shape and following its evolution.

An algorithm can be used to detect the moment the shape splits in two and then construct parameterizations for the two newly obtained curves.

On the bottom row, however, the plane at which the level set function is sampled is translated upwards, on which the shape's change in topology is described.

Thus, in two dimensions, the level-set method amounts to representing a closed curve

, then by chain rule and implicit differentiation, it can be determined that the level-set function

is the Euclidean norm (denoted customarily by single bars in partial differential equations), and

[2][3] However, the numerical solution of the level set equation may require advanced techniques.

If an initial distance field is constructed (i.e. a function whose value is the signed Euclidean distance to the boundary, positive interior, negative exterior) on the initial circle, the normalized gradient of this field will be the circle normal.

This is due to this being effectively the temporal integration of the Eikonal equation with a fixed front velocity.

The level-set method was developed in 1979 by Alain Dervieux,[5] and subsequently popularized by Stanley Osher and James Sethian.